Page 384 - Bird R.B. Transport phenomena

366 Chapter 11 The Equations of Change for Nonisothermal Systems

Fig. 11B.9. Body formed from the intersection of two
cones and a sphere.
Conical
surfaces

Spherical
surface
(insulated)

(11B.8-2)

(a) What are the partial differential equations that must be satisfied by Eqs. 11B.8-1 and 2?
(b) Write down the boundary conditions that apply at r = R.
(c) Show that 7^ and T satisfy their respective partial differential equations in (a).
o
(d) Show that Eqs. 11B.8-1 and 2 satisfy the boundary conditions in (b).
11B.9. Heat flow in a solid bounded by two conical surfaces (Fig. 11B.9). A solid object has the
shape depicted in the figure. The conical surfaces в л = constant and 0 2 = constant are held at
temperatures T and T , respectively. The spherical surface at r = R is insulated. For steady-
2
}
state heat conduction, find
(a) The partial differential equation that T(0) must satisfy.
(b) The solution to the differential equation in (a) containing two constants of integration.
(c) Expressions for the constants of integration.
(d) The expression for the 0-component of the heat flux vector.
(e) The total heat flow (cal/sec) across the conical surface at в = 6
V
- T 2 )
2<TrRk{T x
Answer: (e) Q =
tan
11B.10. Freezing of a spherical drop (Fig. 11B.10). To evaluate the performance of an atomizing noz-
zle, it is proposed to atomize a nonvolatile liquid wax into a stream of cool air. The atomized
wax particles are expected to solidify in the air, from which they may later be collected and

Temperature, T

Fig. 11B.10. Temperature profile in the freez-
ing of a spherical drop.